Due to its favourable properties with regards to the system's total energy and potential vorticity budgets [1]}, the vector-invariant form (REF ) of the momentum equation is popular in NWP models (e.g. [2]}, [3]}).
It has also been used in the context of the compatible finite element
discretisations of the shallow-water equations with vorticity as an auxiliary variable by
[4]}, [5]}, [6]}, and [7]}.
In [6]}, a modified vorticity is diagnosed from the velocity field and used in its transport.
The modification to the vorticity improves the stability of the vector transport term by dissipating enstrophy without compromising on energy conservation.
This is known as the anticipated potential vorticity method (APVM).
However, this approach is not consistent with the continuous equations, in the sense that strong solutions do not necessarily satisfy the discrete equations. To overcome this problem, an extension to the APVM term was developed by [4]}, which used a mixed finite element problem to solve simultaneously for the velocity and the vorticity evolution equations, where the latter is stabilised by an SUPG method. For a recent comparison including the APVM and SUPG methods, see [10]}.
Finally, [5]} introduced a methodology to apply an SUPG-based stabilisation method to more general vorticity evolution equations, which may contain additional terms such as ones arising from temperature gradients.
The methodology relies on the SUPG method's residual-based form.
In this section, we present a mixed velocity-vorticity approach for the vector transport equation similar to the setup of [4]}.
However, here we follow a “black box” approach in which the non-transport terms in the broader equation set are not included in the vorticity transport equation. To achieve this, we apply a residual-based setup akin to the one presented in [5]}.
